Surjection operations provide an explicit and computable tool for expressing various operations from chains on a topological space X to  tensor powers of such chains. For example, this gives a procedure for computing Steenrod operations. In ongoing work with M. Levy, we lift surjection operations to chains in Cartesian powers of X. As an application, we construct an explicit model computing homology of the graph configuration space, obtained by removing from the Cartesian power of X a collection of its diagonals (but possibly not all of them). In particular, this proves an earlier conjecture from a previous joint work with M. Zubkov. We also conjecture that a similar model can be applied to computing factorization homology for knots.